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I thought of this question today: given are a decidable set $A$ and undecidable set $B$ for which $A \cap B = \emptyset$. Is $A \cup B$ decidable or undecidable?

I am almost sure that it is undecidable but cannot come up with an argument/proof as to why, or even a counterexample.

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Hint: Suppose you could decide $A \cup B$. Using an algorithm for deciding $A$ and an algorithm for deciding $A \cup B$, can you decide $B = (A \cup B) \setminus A$?

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